26VIT1984-2010

Creating Stars

26VIT1984-2010

Creating Stars

A Place to Learn ; A Chance To Grow

Arithmetical properties of Tree Generation

Codes and Algorithm to generate all Tree

Codes for a given number of Edges

N.Chandramowliswaran

Applied Algebra Division

School of Advanced Sciences

VIT University

26VIT1984-2010

Creating Stars

ncmowli@hotmail.com

N.Chandramowliswaran

Applied Algebra Division

School of Advanced Sciences

VIT University

26VIT1984-2010

Creating Stars

ncmowli@hotmail.com

Claude Berge Bill Tutte

Generation of Graceful Trees through

Graceful Codes

Abstract

Graceful Code is a way to represent graceful graph in terms

of sequence of non-negative integers. Given a graceful

graph G on “q” edges, we can generate its graceful code in

the form of (a1, a2, a3, …., aq-1, aq=0) to represent the

graph. Similarly, we can easily draw the graph from the

given graceful code.

Graceful codes are classified into two categories, namely,

α-valuable code and gracious code based on their

properties. Graceful code provides an useful and efficient

techniques to study and analyze graphs using computer.

Here we discuss generation of infinitely many graceful

codes, α-valuable codes and gracious codes for a given

graceful code, α-valuable code and a gracious code.

Introduction

A simple graph G(V,E) on “p” vertices and “q”

edges is said to be graceful if there exist

an injection f: V→{0, 1, 2,….,q} such that the

induced function g: E→{1, 2, 3, …, q} which is

defined by g(u, v)=|f(u)-f(v)| for every edge

(u, v), is a bijective function; then “f” is called

graceful labelling of G.

Graceful Code Let G be any graceful graph on “q” edges then

(a1, a2, a3, …, aq -1, aq) is called a graceful code of G,

if 0 ≤ ai ≤ q - i; 1 ≤ i ≤ q.

Here ai is the lower end vertex of the edge label “i”.

It is important to note that aq is always zero

For every graceful graph G we can write its code. Conversely, for every given graceful code we can draw the corresponding graceful graph as follows.

Join edges:(a1,1+a1),(a2,2+a2),...,(aq - 1,q-1+aq-1), (aq, q+aq)

Example 1

Figure 1 shows a graceful graph on q = 7 edges

with edge labeled from 1 to 7.

7

0

6 3

1 2

4 5

Code = (4, 2, 3, 0, 1, 0, 0)

Figure 1

- valuable Code

A graceful code (a1, a2, a3,..., aq-1, aq) of a

graceful graph G on “q” edges is called

α - valuable code if

Here a1 is called the separator or critical value

of the - valuable code.

a1 ai

Max{ai| 1≤ i ≤ q} < Min{i+ai| 1≤ i ≤ q}.

Proposition

(a1, a2, a3, …,aq-1,aq) represents -valuable code if

and only if

0 (a1 – aq - i + 1) / q – i) ≤ 1

for all i , 1 ≤ i ≤ q - 1

Equivalently (a1, a2, a3, …,aq-1,aq) represents an

- valuable code if and only if

(a1 - aq, a1 - aq -1, …, a1 - a3, a1 - a2, 0) represents a

code of a Graceful Graph.

Properties of Graceful Codes

1.1 If (a1, a2, a3,…, aq - 1, aq) represents a code of

a graceful graph G on “q” edges,

then, (a2, a3,…, aq - 1, aq) represents a code

of some graceful graph H on “q - 1” edges.

1.2 If (a1, a2, a3,…, aq-1, aq) is an α–valuable

code on “q” edges and (q -1 - a1 > a1)

then (q – 1 - a1, q – 2 - a2, …, 1- aq - 1, 0,

a1, a2,a3,…, aq) is an α–valuable code

on “2q” edges.

1.3. If (a1, a2, a3,…, aq - 1, aq) is an α–valuable

code on “q” edges and (a1> q -1 - a1) then,

(a1, a2, a3,…, aq, q - 1- a1, q – 2 - a2, …,1 – aq - 1, 0)

is an α–valuable code on “2q” edges.

1.4. If (a1, a2, a3,…, aq1 - 1, aq1) and

(b1, b2, b3, …, bq2 - 1, bq2) represents

α–valuable codes on “q1” and “q2” edges

respectively and a1 ≥ b1 then,

(a1, a2, a3,…, aq1 - 1, aq1, b1, b2, b3,…, bq2 - 1, bq2)

represents an α–valuable code

on “q1 + q2” edges.

1.5. Let (a1, a2, a3,…, aq -1, aq) represents

a graceful code of a graph G on

“q” edges then,

(aq+ q, aq - 1+ q - 1,…, 2 + a2, 1 + a1, a1, a2, a3,…, aq - 1, aq)

represents a α–valuable code on “2q” edges.

Properties of Graceful Codes

If (a1, a2, a3,…, aq - 1, aq) represents a graceful

code of a graceful graph G on “q” edges then,

(aq+ q, aq – 1 + q - 1,…,2 + a2, 1 + a1, x, a1, a2, a3…, aq - 1, aq),

[0 ≤ x ≤ q] represents an α–valuable code

on “2q + 1” edges.

If (a1, a2, …, aq 1, aq) represent a code of a

graceful graph G on q edges,

Then,

(q – aq, q – aq 1, …, q – a2, q – a1, a1, a2, …, aq 1, aq)

represent a -valuable code on “2 q” edges.

Properties of Graceful Codes

Let X1, X2, X3, …, Xr represent “r” α–valuable

codes on edges “qi”

(1 i r) having separators “si” respectively.

Then, r-1 r-2 r-3

( sj + Xr , sj + Xr-1 , sj + Xr-2 , … s1+ s2+ X3, s1+ X2, X1)

j=1 j=1 j=1

r

always represent a α–valuable code on qj edges.

j = 1

Tree Generation Theorems

Let G be any simple graph on “n” vertices and “q”

edges.

Define a bipartite graph HG as follows:

(vi, vj) E(G) <=> (vi, vj’) E(HG) and

(vi’, vj) E(HG).

Join any vk V(G) V(HG),

[1 ≤ k ≤ n] to vk’ V(HG).

Here |V(HG)| = 2|V(G)| and |E(HG)| = 2 | E(G)| +1.

Tree Generation Theorems

Moreover if G has a code (a1, a2, a3,…, aq - 1, aq)

then HG has an α–valuable code

(aq+ q, aq-1+q - 1,…, 2+a2, 1 + a1, x, a1, a2, a3, …, aq - 1, aq) [0 ≤ x ≤ q].

If G happens to be a bipartite graph, then HG

contains two copies of G

together with an edge connecting vk to vk’

Examples

Code = (0, 1, 0, 0)

G

HG

Code = (4, 3, 3, 1, 3, 0, 1, 0, 0)

Examples

G

Construction of HG

ai i + ai

i

E (G)

ai i + ai

q+1+ai q+1+i+ai

E(HG)

q+1+i q+1-i

(aq+q, …,ai+i, …,1+a1, x, a1, …, ai, …, aq)

q+1+ai q+1+i+ai

q+1-i q+1+i

i i

Tree Generation Theorems

Theorem

If (a1, a2, a3,…, aq -1, aq) represents a α–valuable

code of some tree “T” . Then,

(aq+q, aq - 1+q-1, …,2+a2, 1+a1, a1, a2, a3,…, aq - 1, aq)

represents a α–valuable code of a tree “S”

on “2q” edges such that

E(S) = E(T) U E(T).

Tree Generation Theorems

Theorem

If (a1, a2, a3,…, aq-1, aq) is an α–valuable code of a

graceful graph G on “q” edges, then,

(a1, a2, a3,…, aq-1, aq) represents a tree if and only if

(a2, a3,…, aq -1, aq) represents a tree

on “q - 1” edges.

Tree Generation Theorems

If (a1, a2, …, aq-2, aq-1, aq ) represents a code of a

graceful tree on ‘q’ edges, then

1. (qk - 1, ka1, (q –1) k –1, ka2, …, 2 k - 1, kaq - 1, 1k - 1, kaq)

represent a tree code on “kq” edges (k 2).

2. (qk - 1, ka1+r, (q –1) k – 1, ka2+r, …, 2k - 1, kaq-1+r, 1k - 1,

kaq+r, 0r) ;1 ≤ r ≤ k, k ≥ 2 represent a tree code

on “kq+r” edges.

Corollary – 1

If (a1, a2, …, aq - 2, aq - 1, aq ) represents a code of a

graceful tree on ‘q’ edges, then

(q, 2a1, q - 1, 2a2, q - 2, 2a3, …, 2, 2aq - 1, 1, 2aq)

represent a code of a graceful tree on “2q” edges and (q, 2a1+1, q - 1, 2a2+1, q - 2, 2a3+1, …, 2, 2aq - 1+1, 1,

2aq+1,0) represent a tree code on “2q+1” edges.

Corollary – 2

If (a1, a2, …, aq-2, aq-1, aq ) represents a code of a

graceful tree on ‘q’ edges, then

(q+1, 2a1, q, 2a2, q - 1, 2a3, …, 3, 2aq-1, 2, 2aq, 1, 0)

represent a code of a graceful tree on “

2q+2” edges and (q+1, 2a1+1, q, 2a2+1, q - 1,

2a3+1, …, 3, 2aq - 1+1, 2, 2aq+1, 1, 0, 0)

represent a tree code on “2q + 3” edges.

Tree Generation Theorems

Using - valuable tree codes

Theorem 1

If (a1, a2, …, aq - 1, aq) represent a -valuable tree

code on “q” edges, then,

(aq+q, aq - 1+ q – 1, …, 2 + a2, 1+ a1, 1 + a1, a1, a1,a2,

…, aq - 1, aq)

represent a -valuable tree code on “2q+2” edges.

Theorem 2

Let (a1, a2, …, aq1 - 2, aq1 - 1, aq1) represents a

- valuable tree code on “q1” edges and

(b1, b2, …, bq2 - 2, bq2 - 1, bq2) represent a tree code on

“q2” edges. Then,

1. (a1 + b1 , a1 + b2, …, a1 + bq2 - 2, a1+ bq2 - 1, a1+ bq2,

a1, a2, …, aq1 - 2, aq1 - 1, aq1 ) represent a tree code on “q1 + q2” edges.

Tree Generation Theorems Using

- valuable tree codes

2. (a1+ b1, a1+b2, …, a1+ bq2 - 2, a1+bq2 - 1, a1+bq2, a1 – aq1, a1 – aq1 - 1,a1 aq1 - 2, …, a1 – a2, 0)

represent a tree code on “q1+ q2” edges.

3. (q1– 1 – a1 + b1, q1– 1 – a1+ b2, q1– 1 – a1+ b3,

…, q1– 1– a1+ bq2 - 2, q1– 1– a1+ bq2 - 1, q1– 1 –

a1+ bq2, q1– 1– a1, q1 – 2 – a2, …, 2 aq1 - 2, 1

aq1 - 1, 0) represent a tree code on “q1+ q2” edges.

Tree Generation Theorems Using

- valuable tree codes

4. (q1– 1 – a1+ b1, q1– 1 – a1+ b2, q1– 1 – a1+ b3 , …, q1– 1 – a1+ bq2 - 2, q1– 1 – a1+ bq2 - 1, q1– 1 – a1+ bq2, q1– 1 – a1, q1– 2 – (a1 aq1 - 1),

q1 – 3 – (a1 aq1 - 2), …, 1 – (a1 – a2), 0) represent a tree code on “q1+ q2” edges.

5. (a1+ a2, a1+ a3, a1+ a4, …, a1+ aq1 - 2, a1+ aq1 - 1,

a1+ aq1, a1, a2, …, aq1 - 2, aq1-1, aq1) represent a

tree code on “2q – 1” edges.

Tree Generation Theorems Using

- valuable tree codes

Corollary 1

Let X1, X2, X3, …, Xr represent “r” α–valuable tree codes on edges “qi”

(1 i r) having separators “si” respectively.

Then

r- 1 r- 2 r- 3

( sj + Xr, sj + Xr - 1, sj + Xr - 2, …, s1+ s2+ X3, s1+ X2, X1)

j=1 j=1 j=1

r

always represent a α–valuable tree code on qj edges.

j=1